Question: Simplify the following expression and state the condition under which the simplification is valid. You can assume that $x \neq 0$. $k = \dfrac{4x}{3(3x + 1)} \div \dfrac{-4}{10(3x + 1)} $
Explanation: Dividing by an expression is the same as multiplying by its inverse. $k = \dfrac{4x}{3(3x + 1)} \times \dfrac{10(3x + 1)}{-4} $ When multiplying fractions, we multiply the numerators and the denominators. $k = \dfrac{ 4x \times 10(3x + 1) } { 3(3x + 1) \times -4 } $ $ k = \dfrac{40x(3x + 1)}{-12(3x + 1)} $ We can cancel the $3x + 1$ so long as $3x + 1 \neq 0$ Therefore $x \neq -\dfrac{1}{3}$ $k = \dfrac{40x \cancel{(3x + 1})}{-12 \cancel{(3x + 1)}} = -\dfrac{40x}{12} = -\dfrac{10x}{3} $